周三. 7月 17th, 2019

YueTa

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三分钟记住和差化积公式

和差化积公式

 

即三角函数中的一组恒等式: [1]

推导过程

对于(1)至(4),可用积化和差公式推导,也可以由和角公式得到,以下用和角公式证明之。
由和角公式有,
两式相加、减便可得到上面的公式(1)、(2),同理可证明公式(3)、(4)。
对于(5)、(6),有
对于(7)、(8)、(9)、(10),也可用类似的方法推出。
证毕。

平方形式的和差化积公式

编辑

下面不加推导地给出几个公式。对于正余弦平方的减法,同样有和差化积公式: [1]

记忆方法

编辑

和差化积公式的形式比较复杂,记忆中以下几个方面是难点,下面指出了各自的简单记忆方法。

如何只记两个公式甚至一个

我们可以只记上面四个公式的第一个和第三个。

第二个公式中的

 

,即

 

,这就可以用第一个公式。

同理,第四个公式中,

 

,这就可以用第三个公式解决。

如果对诱导公式足够熟悉,可以在运算时把余弦全部转化为正弦,那样就只记住第一个公式就行了。
用的时候想得起一两个就行了。

结果乘以2

这一点最简单的记忆方法是通过三角函数的值域判断。正弦和余弦的值域都是[-1,1],其积的值域也应该是[-1,1],而和差的值域却是[-2,2] ,因此乘以2是必须的。
也可以通过其证明来记忆,因为展开两角和差公式后,未抵消的两项相同而造成有系数2,如:
故最后需要乘以2。

只有同名三角函数能和差化积

无论是正弦函数还是余弦函数,都只有同名三角函数的和差能够化为乘积。这一点主要是根据证明记忆,因为如果不是同名三角函数,两角和差公式展开后乘积项的形式都不同,就不会出现相抵消和相同的项,也就无法化简下去了。

记忆口诀

(一)

正加正,正在前,

 

余加余,余并肩。

 

正减正,余在前,

 

余减余,负正弦。

 
(反之亦然)
(二)

帅+帅=帅哥,

 

帅-帅=哥帅,

 

哥+哥=哥哥,

 

哥-哥=负嫂嫂。

 
(反之亦然)
(三)

口口之和仍口口,

 

赛赛之和赛口留,

 

口口之差负赛赛,

 

赛赛之差口赛收。

 
(四)

正和正在先,

 

正差正后迁,

 

余和一色余,

 

余差翻了天。

 
(五)

正弦加正弦,正弦在前面,

 

正弦减正弦,余弦在前面,

 

余弦加余弦,余弦全部见,

 

余弦减余弦,负正弦来见。

 

 

 

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